The notion of robustness can be described as follows. Suppose that the plant transfer function P belongs to a set P, as in the preceding section. Consider some characteristic of the feedback system, for example, that it is internally stable. A controller C is robust with respect to this characteristic if this characteristic holds for every plant in P. The notion of robustness therefore requires a controller, a set of plants, and some characteristic of the system. For us, the two most important variations of this notion are robust stability, treated in this section, and robust performance, treated in the next.
A controller C provides robust stability if it provides internal stability for every plant in P. We might like to have a test for robust stability, a test involving C and P. Or if P has an associated size, the maximum size such that C stabilizes all of P might be a useful notion of stability margin. The Nyquist plot gives information about stability margin. Note that the distance from the critical point -1 to the nearest point on the Nyquist plot of L equals 1/kSk∞:
distance from -1 to Nyquist plot = inf
ω | − 1 − L(jω)| = inf ω |1 + L(jω)| = sup ω 1 |1 + L(jω)| −1 = kSk−1∞.
Thus if kSk∞ ≫ 1, the Nyquist plot comes close to the critical point, and the feedback system is
nearly unstable. However, as a measure of stability margin this distance is not entirely adequate because it contains no frequency information. More precisely, if the nominal plant P is perturbed to ˜P , having the same number of unstable poles as has P and satisfying the inequality
| ˜P (jω)C(jω) − P (jω)C(jω)| < kSk−1∞, ∀ω,
then internal stability is preserved (the number of encirclements of the critical point by the Nyquist plot does not change). But this is usually very conservative; for instance, larger perturbations could be allowed at frequencies where P (jω)C(jω) is far from the critical point.
Better stability margins are obtained by taking explicit frequency-dependent perturbation mod- els: for example, the multiplicative perturbation model, ˜P = (1 + ∆W2)P . Fix a positive number
β and consider the family of plants { ˜P : ∆ is stable and k∆k∞ ≤ β}.
Now a controller C that achieves internal stability for the nominal plant P will stabilize this entire family if β is small enough. Denote by βsupthe least upper bound on β such that C achieves internal
stability for the entire family. Then βsup is a stability margin (with respect to this uncertainty
model). Analogous stability margins could be defined for the other uncertainty models.
We turn now to two classical measures of stability margin, gain and phase margin. Assume that the feedback system is internally stable with plant P and controller C. Now perturb the plant to kP , with k a positive real number. The upper gain margin, denoted kmax, is the first value of
k greater than 1 when the feedback system is not internally stable; that is, kmax is the maximum
4.2. ROBUST STABILITY 51 kmax := ∞. Similarly, the lower gain margin, kmin, is the least nonnegative number such that
internal stability holds for kmin< k ≤ 1. These two numbers can be read off the Nyquist plot of L;
for example, −1/kmaxis the point where the Nyquist plot intersects the segment (−1, 0) of the real
axis, the closest point to −1 if there are several points of intersection.
Now perturb the plant to e−jφP , with φ a positive real number. The phase margin, φmax, is the
maximum number (usually expressed in degrees) such that internal stability holds for 0 ≤ φ < φmax.
You can see that φmax is the angle through which the Nyquist plot must be rotated until it passes
through the critical point, −1; or, in radians, φmax equals the arc length along the unit circle from
the Nyquist plot to the critical point.
Thus gain and phase margins measure the distance from the critical point to the Nyquist plot in certain specific directions. Gain and phase margins have traditionally been important measures of stability robustness: if either is small, the system is close to instability. Notice, however, that the gain and phase margins can be relatively large and yet the Nyquist plot of L can pass close to the critical point; that is, simultaneous small changes in gain and phase could cause internal instability. We return to these margins in Chapter 11.
Now we look at a typical robust stability test, one for the multiplicative perturbation model. Assume that the nominal feedback system (i.e., with ∆ = 0) is internally stable for controller C. Bring in again the complementary sensitivity function
T = 1 − S = L 1 + L =
P C 1 + P C.
Theorem 1 (Multiplicative uncertainty model) C provides robust stability iff kW2T k∞< 1.
Proof (⇐) Assume that kW2T k∞ < 1. Construct the Nyquist plot of L, indenting D to the left
around poles on the imaginary axis. Since the nominal feedback system is internally stable, we know this from the Nyquist criterion: The Nyquist plot of L does not pass through -1 and its number of counterclockwise encirclements equals the number of poles of P in Res ≥ 0 plus the number of poles of C in Res ≥ 0.
Fix an allowable ∆. Construct the Nyquist plot of ˜P C = (1+∆W2)L. No additional indentations
are required since ∆W2 introduces no additional imaginary axis poles. We have to show that
the Nyquist plot of (1 + ∆W2)L does not pass through -1 and its number of counterclockwise
encirclements equals the number of poles of (1 + ∆W2)P in Re s ≥ 0 plus the number of poles of C
in Re s ≥ 0; equivalently, the Nyquist plot of (1 + ∆W2)L does not pass through -1 and encircles
it exactly as many times as does the Nyquist plot of L. We must show, in other words, that the perturbation does not change the number of encirclements.
The key equation is
1 + (1 + ∆W2)L = (1 + L)(1 + ∆W2T ). (4.1)
Since
k∆W2T k∞ ≤ kW2T k∞< 1,
the point 1 + ∆W2T always lies in some closed disk with center 1, radius < 1, for all points s on D.
Thus from (4.1), as s goes once around D, the net change in the angle of 1 + (1 + ∆W2)L equals
(⇒) Suppose that kW2T k∞≥ 1. We will construct an allowable ∆ that destabilizes the feedback
system. Since T is strictly proper, at some frequency ω,
|W2(jω)T (jω)| = 1. (4.2)
Suppose that ω = 0. Then W2(0)T (0) is a real number, either +1 or −1. If ∆ = −W2(0)T (0), then
∆ is allowable and
1 + ∆W2(0)T (0) = 0.
From (4.1) the Nyquist plot of (1 + ∆W2)L passes through the critical point, so the perturbed
feedback system is not internally stable.
If ω > 0, constructing an admissible ∆ takes a little more work; the details are omitted. The theorem can be used effectively to find the stability margin βsup defined previously. The
simple scaling technique
{ ˜P = (1 + ∆W2)P : k∆k∞≤ β} = { ˜P = (1 + β−1∆βW2)P : kβ−1∆k∞≤ 1}
= { ˜P = (1 + ∆1βW2)P : k∆1k∞≤ 1}
together with the theorem shows that
βsup = sup{β : kβW2T k∞< 1} = 1/kW2T k∞.
The condition kW2T k∞< 1 also has a nice graphical interpretation. Note that
kW2T k∞< 1 ⇔ W2(jω)L(jω) 1 + L(jω) < 1, ∀ω ⇔ |W2(jω)L(jω)| < |1 + L(jω)|, ∀ω.
The last inequality says that at every frequency, the critical point, -1, lies outside the disk of center L(jω), radius |W2(jω)L(jω)| (Figure 4.3). &% '$ r r - |W2L| L −1
Figure 4.3: Robust stability graphically.
There is a simple way to see the relevance of the condition kW2T k∞< 1. First, draw the block
diagram of the perturbed feedback system, but ignoring inputs (Figure 4.4). The transfer function from the output of ∆ around to the input of ∆ equals −W2T , so the block diagram collapses to
the configuration shown in Figure 4.5. The maximum loop gain in Figure 4.5 equals k − ∆W2T k∞,
which is < 1 for all allowable ∆s iff the small-gain condition kW2T k∞< 1 holds.
The foregoing discussion is related to the small-gain theorem, a special case of which is this: If L is stable and kLk∞< 1, then (1 + L)−1 is stable too. An easy proof uses the Nyquist criterion.
4.3. ROBUST PERFORMANCE 53